{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Discrete Random Variable**\n",
    "\n",
    "A **random variable** is a real-valued function of a probabilistic sample space. It is **discrete** if it's range is *finite* or countably infinite, i.e. if a one-to-one correspondance can be created between it's range and the positive integers.\n",
    "\n",
    "A discrete random variable has an associated **probability mass function** (PMF) (aka probability distribution function) that gives the probability of each range value for the random variable. If *x* is any real number, the **probability mass** of *x* is the probability of the event $\\left\\{X=x\\right\\}\\hspace{1pt}$ consisting of all outcomes that give rise to the value *x* for *X* and is denoted\n",
    "\n",
    "$$p_X\\left(x\\right) = P\\left(\\left\\{X=x\\right\\}\\right)$$\n",
    "\n",
    "The PMF of a random variable *X* can be calculated as follows:\n",
    "\n",
    "For each possible value *x* of *X*:\n",
    "\n",
    "1. Collect all possible disjoint events in $\\Omega\\hspace{1pt}$ that give rise to the event $\\left\\{X=x\\right\\}\\hspace{1pt}$\n",
    "\n",
    "2. Add their probabilities to obtain $p_X\\left(x\\right)$\n",
    "\n",
    "3. Repeat for all *x*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Functions of Random Variables**\n",
    "\n",
    "If $Y=g\\left(X\\right)\\hspace{1pt}$ is a function of a random variable *X*, then *Y* is also a random variable. If *X* is discrete with PMF $p_X\\hspace{1pt}$, then *Y* is also discrete and its PMF is \n",
    "\n",
    "$$p_Y\\left(y\\right) = \\sum_{\\{x \\hspace{1pt}\\vert \\hspace{1pt} g\\left(x\\right)=y\\}} p_x\\left(x\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Expectation (Mean)**\n",
    "\n",
    "The **expected** value, or mean, of a random variable *X* with PMF $p_X\\hspace{1pt}$ is defined as\n",
    "\n",
    "$$E\\left[X\\right] = \\sum_x x p_X\\left(x\\right)$$\n",
    "\n",
    "Note that the expectation is well-defined only if \n",
    "\n",
    "$$\\sum_x \\left| x \\right| p_X \\left(x\\right) \\lt \\infty$$\n",
    "\n",
    "This fails to hold for certain distributions. For example, consider a power law PMF of the form $p_X\\left(x\\right) = \\frac{1}{x^{\\alpha+1}}\\hspace{1pt}$ with $\\alpha\\le1$ for $x$ any positive integer. Then the expected value,$\\sum_{n=1}^{\\infty}\\frac{1}{x^{\\alpha}}$ is a divergent <a href=https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#p-series>p-series</a>.\n",
    "\n",
    "**Expected Value of Functions of Random Variables**\n",
    "\n",
    "Let *X* be a random variable with PMF $p_X\\hspace{1pt}$ and let $g\\left(X\\right)\\hspace{1pt}$ be a function of *X*. Then the expected value of the random variable $Y=g\\left(X\\right)\\hspace{1pt}$ is \n",
    "\n",
    "$$E\\left[Y\\right]=E\\left[g\\left(X\\right)\\right] = \\sum_x g\\left(x\\right) p_X \\left(x\\right)$$\n",
    "\n",
    "**Note** that unless $g\\left(X\\right)\\hspace{1pt}$ is a linear function, it is **not** generally true that $E\\left[g\\left(X\\right)\\right] = g\\left(E\\left[X\\right]\\right)\\hspace{1pt}$. This is illustrated in the following example.\n",
    "\n",
    "*Example*\n",
    "Problem 21 page 123 of [1]. Assume you toss a fair coin until the first tails appears at say toss *n* and that you receive $2^n\\hspace{1pt}$ dollars as a reward. What is the expected about of your winnings? How much would you pay to play the game?\n",
    "\n",
    "Let the RV *X* be the amount you receive as a reward. Let the RV *N* be the number of tosses up to an including the first tail. Then we have $X=g\\left(N\\right)=2^N\\hspace{1pt}$. We know that *N* can be represented as a geometric random variable with PMF, with $p=\\frac{1}{2}$ assuming a fair coin\n",
    "\n",
    "$$P_N\\left(n\\right) = \\left(\\frac{1}{2}\\right)^n$$\n",
    "\n",
    "with expected value\n",
    "\n",
    "$$E\\left[N\\right] = \\sum_{n=1}^{\\infty}n\\left(\\frac{1}{2}\\right)^n = \\frac{1}{p} = 2$$\n",
    "\n",
    "and it is *tempting, but wrong*, to argue that the expected reward is $2^{E\\left[N\\right]}=2^2=4\\hspace{1pt}$. But such an argument assumes that \n",
    "\n",
    "$$E\\left[g\\left(N\\right)\\right]=E\\left[2^N\\right]=2^{E\\left[N\\right]}=g\\left(E\\left[N\\right]\\right)$$\n",
    "\n",
    "which is not true. Consider again the RV X. It takes on values $2^0, 2^1, 2^2, \\ldots\\hspace{1pt}$, i.e. values $2^n\\hspace{1pt}$ for any integer *n*. Noting that a particular outcome occurs with the same probability as obtaining the first tails on the $n^{th}$ toss, the expected value is \n",
    "\n",
    "$$E\\left[X\\right] = \\sum_{k=0}^{\\infty}2^k 2^{-k} = \\infty$$\n",
    "\n",
    "So it is argued that the expected, or mean payout is infinite, implying that one should pay **any** amount to play this game. However, it is **more accurate to state that the expected value is undefined** and so too is the variance. Several researchers have presented alternative risk assesments for determining a fair *entrance* fee for this game - try a quick Google search. Below are simulated outcomes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Max payout = 16384\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "games = 10000\n",
    "p = 0.5 #use a fair coin\n",
    "\n",
    "def playGame(p, n=0):\n",
    "    r = numpy.random.rand()\n",
    "    if r<=p: return n\n",
    "    else: return playGame(p,n+1)\n",
    "    \n",
    "payouts = [2**playGame(p=p, n=0) for i in range(games)]\n",
    "cumAvg = numpy.cumsum(payouts)\n",
    "cumAvg = [cumAvg[i]/float((i+1)) for i in range(games)]\n",
    "\n",
    "f,axarr = pyplot.subplots(1,2)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "axarr[0].scatter(range(games),payouts)\n",
    "axarr[1].plot(cumAvg)\n",
    "\n",
    "print \"Max payout = {0}\".format(max(payouts))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Higher Order Moments**\n",
    "\n",
    "The expected value is special case of the more general notion of the $n^{th}$ **moment** of a random variable $X$, defined as\n",
    "\n",
    "$$E\\left[X\\right] = \\sum_x x^n p_X\\left(x\\right)$$\n",
    "\n",
    "where again this is only well defined if \n",
    "\n",
    "$$\\sum_x \\left| x^n \\right| p_X \\left(x\\right) \\lt \\infty$$\n",
    "\n",
    "As with the mean, this condition can fail for certain distributions. Consider the power law PMF described previously. All moments, *m*, such that $m\\ge\\alpha\\hspace{1pt}$ fail to satisfy this condition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Variance**\n",
    "\n",
    "The **variance** of a random variable, denoted $var\\left(X\\right)\\hspace{1pt}$ is defined as the expected value of the random variable $\\left(X - E\\left[X\\right]\\right)^2\\hspace{1pt}$ and the following identities hold\n",
    "\n",
    "$$\\begin{eqnarray}var\\left(X\\right) &=& E\\left[\\left(X-E\\left[X\\right]\\right)^2\\right] \\cr\n",
    "        &=& \\sum_x\\left(x-E\\left[X\\right]\\right)^2 p_X\\left(x\\right) \\cr\n",
    "        &=& E\\left[X^2\\right] - \\left(E\\left[X\\right]\\right)^2\n",
    "\\end{eqnarray}$$\n",
    "\n",
    "The variance is a measure of the expected value of the square of difference between an arbitrary observation of a RV and the mean of that RV.  \n",
    "\n",
    "The **standard deviation**, $\\sigma_X\\hspace{1pt}$, is defined as the square root of the variance.\n",
    "\n",
    "**Linear Function of a Random Variable**\n",
    "\n",
    "Let *X* be a random variable and let $Y=aX+b\\hspace{1pt}$ for scalars *a* and *b*. Then\n",
    "\n",
    "$$E\\left[Y\\right] = a E\\left[X\\right] + b$$\n",
    "\n",
    "and \n",
    "\n",
    "$$var\\left(Y\\right) = a^2var\\left(X\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Joint PMFs of Multiple Random Variables**\n",
    "\n",
    "Consider two discrete random variables *X* and *Y*. The probabilities of the pairs $\\left(x,y\\right)\\hspace{1pt}$ that the *X* and *Y* can take are described by the **joint PMF** denoted $p_{X,Y}$\n",
    "\n",
    "$$p_{X,Y}\\left(x,y\\right) = P\\left(X=x,Y=y\\right)$$\n",
    "\n",
    "From the joint PMF, the PMF of *X* or *Y* can be obtained as\n",
    "\n",
    "$$p_X\\left(x\\right) = \\sum_y p_{X,Y}\\left(x,y\\right) \\quad p_Y\\left(y\\right) = \\sum_x p_{X,Y}\\left(x,y\\right)$$\n",
    "\n",
    "where here $p_X\\hspace{1pt}$ and $p_Y\\hspace{1pt}$ are referred to as the **marginal PMFs**, though their meaning and values are exactly the same as the standard PMF of the single variables $X$ and $Y$. The term *marginal* simply distinguishes them from the full joint PMF.\n",
    "\n",
    "Naturally, these notions can be extended to any number of discrete random variables $X_1, \\ldots, X_n\\hspace{1pt}$ with joint PMF\n",
    "\n",
    "$$p_{X_1,\\ldots,X_n}\\left(x_1,\\ldots,x_n\\right) = P\\left(X_1=x_1,\\ldots,X_n=x_n\\right)$$\n",
    "\n",
    "For any $X_i\\hspace{1pt}$ let $X_1, \\ldots, X_{n-1}$ be the remaining random variables from $X_1, \\ldots, X_n$ the PMF of $X_i\\hspace{1pt}$ is \n",
    "\n",
    "$$p_{X_i}\\left(x\\right) = \\sum_{x_1}\\ldots\\sum_{x_{n-1}}p_{X_1\\ldots,X_n}\\left(x_1,\\ldots,x_{n-1}\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Functions of Multiple Random Variables**\n",
    "A function $Z=g\\left(X,Y\\right)\\hspace{1pt}$ of the random variables *X* and *Y* is a random varibale with PMF\n",
    "\n",
    "$$p_Z\\left(z\\right) = \\sum_{\\left\\{ \\left(x,y\\right) \\vert g\\left(x,y\\right)=z\\right\\}} p_{X,Y}\\left(x,y\\right)$$\n",
    "\n",
    "The expected value of *Z* is\n",
    "\n",
    "$$E\\left[g\\left(X,Y\\right)\\right] = \\sum_x\\sum_y g\\left(x,y\\right) p_{X,Y}\\left(x,y\\right) $$\n",
    "\n",
    "These notions can be extended to any number of discrete random variables. Let $\\bar{X_s} = \\left\\{X_1,\\ldots,X_n\\right\\}\\hspace{1pt}$ be a set of discrete random variables with joint PMF $p_{\\bar{X_s}}\\hspace{1pt}$. Let *Z* be the random variable defined by $Z=g\\left(\\bar{X_s}\\right)\\hspace{1pt}$, then *Z* has PMF\n",
    "\n",
    "$$p_Z(z) = \\sum_{\\left\\{\\left(x_1,\\ldots,x_n\\right) \\hspace{2pt}\\vert \\hspace{2pt} g\\left(x_1,\\ldots,x_n\\right)=z\\right\\}} p_{\\bar{X_s}}\\left(x_1,\\ldots, x_n\\right)$$\n",
    "\n",
    "and the expected value of *Z* is\n",
    "\n",
    "$$E\\left[Z\\right] = E\\left[g\\left(\\bar{X_s}\\right)\\right] = \\sum_{x_1}\\ldots\\sum_{x_n}g\\left(x_1,\\ldots,x_n\\right)p_{\\bar{X_s\n",
    "}}\\left(x_1,\\ldots,x_n\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Conditioning a RV on another RV**\n",
    "\n",
    "Let *X* and *Y* be two random variables. The **conditional PMF** of *X* given *Y* is \n",
    "\n",
    "$$p_{X\\vert Y} \\left(x \\vert y \\right) = \n",
    "\\frac {P\\left( X=x,Y=y \\right)} {P\\left( Y=y \\right)} = \n",
    "\\frac{P \\left( \\left\\{ X=x \\right\\} \\cap \\left\\{ Y=y \\right\\} \\right)} {P\\left( \\left\\{Y=y\\right\\} \\right)}\n",
    "= \\frac{p_{X,Y}\\left(x,y\\right)}{p_Y\\left(y\\right)}$$\n",
    "\n",
    "The conditional probability is itself a valid probability model and satisfies\n",
    "\n",
    "$$\\sum_x p_{X\\vert Y} \\left(x\\vert y\\right) = 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Conditional Expectation**\n",
    "\n",
    "Let *X* and *Y* be random variables. The conditional expectation of *X* given *Y=y* is \n",
    "\n",
    "$$E\\left[X\\vert Y =y\\right] = \\sum_x x p_{X \\vert Y}\\left(x \\vert y\\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Total Expectation Theorem**\n",
    "If $A_1,\\ldots,A_n\\hspace{1pt}$ is a disjoint partition of the sample space with $P\\left(A_i\\right)>0$ for all *i* then\n",
    "\n",
    "$$E[X] = \\sum_i^n P\\left(A_i\\right) E\\left[X\\vert A_i \\right]$$\n",
    "\n",
    "In words, this states that expected value of *X* is a weighted average of the expected values of *X* under the events $A_i\\hspace{1pt}$ where the weights are the probability of the events $A_i\\hspace{1pt}$ provided they form a partition of the sample space.\n",
    "\n",
    "For any event *B* with $P\\left(A_i \\cap B\\right) > 0 \\hspace{1pt}$ for all *i*, we have\n",
    "\n",
    "$$E\\left[X\\vert B \\right] = \\sum_{i=1}^n P \\left( A_i \\vert B \\right) E \\left[X \\vert A_i \\cap B \\right]$$\n",
    "\n",
    "where this has a similar interpretation as a weighted average of the expected values of *X* under the events  $A_i \\cap B\\hspace{1pt}$. Finally, we also have\n",
    "\n",
    "$$ E \\left[X \\right] = \\sum_y p_Y\\left( y \\right) E\\left[X \\vert Y=y \\right]$$\n",
    "\n",
    "which can also be intrepreted as stating that the expected value of *X* is a weighted average of the conditional expectation of *X* conditioned on *Y* where the weights are the PMF of *Y*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "Problem 33 page 128 of [1]. \n",
    "\n",
    "A coin that has probability of heads equal to *p* is tossed successively and independently until a head or tails comes twice in a row. Find the exepcted value of the number of tosses.\n",
    "\n",
    "As shown in the text, the *total expectation theorem* can be used to obtain the solution\n",
    "\n",
    "$$E\\left[X\\right] = \\frac{2+pq}{1-pq}$$\n",
    "\n",
    "where $q=1-p\\hspace{1pt}$. \n",
    "\n",
    "In particular, note that the expected value is bounded, $2\\le E\\left[X\\right]\\le 3\\hspace{1pt}$. This is observed in the simulations below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pValues = [0.2*(i+1) for i in range(3)]\n",
    "numGames = 5000\n",
    "\n",
    "def playGame(p, count = 0, lastToss = -1):\n",
    "    heads = numpy.random.rand()<=p\n",
    "    if lastToss == -1: return playGame(p,count+1, heads)\n",
    "    if (lastToss and heads) or (not lastToss and not heads): return count + 1\n",
    "    else: return playGame(p, count+1, heads)\n",
    "    \n",
    "rslts = []\n",
    "for p in pValues:\n",
    "    rslt = []\n",
    "    for g in range(numGames):\n",
    "        rslt.append(playGame(p))\n",
    "    rslts.append(rslt)\n",
    "\n",
    "cumAvgs = []\n",
    "for r in rslts:\n",
    "    cumSum = numpy.cumsum(r)\n",
    "    cumAvgs.append([cumSum[i]/float((i+1)) for i in range(numGames)])\n",
    "\n",
    "f,axarr = pyplot.subplots(1,2)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "for r in rslts: axarr[0].plot(range(min(100,numGames)),r[0:min(100,numGames)])\n",
    "for ca in cumAvgs: axarr[1].plot(range(numGames), ca)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Independence of Random Variables**\n",
    "\n",
    "Two random variables *X* and *Y* are **independent** if\n",
    "\n",
    "$$p_{X,Y} \\left(x,y\\right) = p_X \\left(x\\right) p_Y \\left(y\\right) \\quad \\forall \\hspace{2pt} x,y$$\n",
    "\n",
    "which is equivalent to requiring that the events $\\left\\{X=x\\right\\}\\hspace{1pt}$ and $\\left\\{Y=y\\right\\}\\hspace{1pt}$ are independent **for all** combinations of *x* and *y*. This concept can be extended to any number of random variables, that is the set of random variables $X_1,\\ldots,X_n\\hspace{1pt}$ are independent if\n",
    "\n",
    "$$P_{X_1,\\ldots,X_n} \\left(x_1,\\ldots,x_n\\right) = p_{X_1}\\left(x_1\\right) \\ldots p_{X_n}\\left(x_n\\right) \\quad \\forall \\hspace{2pt} x_1,\\ldots,x_n$$\n",
    "\n",
    "Notice that unlike the case of independent **events** we do not explicitly require the sub-conditions of independence of all the different combinations of the random variables. In fact, these conditions are embedded in the given condition because it must hold for all $x_1,\\ldots,x_n\\hspace{1pt}$.\n",
    "\n",
    "*X* and *Y* are said to **conditionally independent** given an event *A* if\n",
    "\n",
    "$$p_{X,Y \\vert A} \\left(x,y\\right) = p_{X\\vert A}\\left(x\\right) p_{Y \\vert A}\\left(y\\right) \\quad \\forall \\hspace{2pt} x,y$$\n",
    "\n",
    "If $X_1,\\ldots,X_n$ are independent\n",
    "\n",
    "$$var\\left(X_1 + \\ldots + X_n\\right) = var\\left(X_1\\right) + \\ldots var\\left(X_n\\right)$$\n",
    "\n",
    "i.e. the variance of the sum of **independent** random variables is equal to the sum of the variances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   "outputs": [],
   "source": []
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